If $a$ and $b$ are natural numbers, then $a-b$ is an integer and so the square $(a-b)^2$ is a natural number. In particular

$$ (a-b)^2 \geq 0. \qquad (1)$$

Combining this fact with the identity

$$ ab + ba + (a-b)^2 = a^2 + b^2 \qquad (2)$$

we obtain the inequality

$$ ab + ba \leq a^2 + b^2 \qquad (3)$$

which can be viewed as a special case of either the arithmetic mean-geometric mean inequality or the Cauchy-Schwarz inequality.

Of course this argument can be generalised; for instance, if $v, w$ are elements of a (real or complex) inner product space, then

$$ \langle v-w, v -w \rangle \geq 0 \qquad (1')$$

and by combining this with the identity

$$ \langle v, w \rangle + \langle w, v \rangle + \langle v-w, v-w \rangle = \langle v,v \rangle + \langle w, w \rangle \qquad (2')$$

we conclude the inequality

$$ \langle v, w \rangle + \langle w, v \rangle \leq \langle v, v \rangle + \langle w, w \rangle \qquad (3')$$

which is closely related to the Cauchy-Schwarz inequality (indeed one can amplify (3') to the Cauchy-Schwarz inequality, as discussed in this blog post of mine).

The inequalities (3) or (3') can be interpreted in various categories. For instance, (3) in the category of finite sets becomes

Theorem 1. Let $A, B$ be finite sets. Then there exists an injection from $(A \times B) \uplus (B \times A)$ to $(A \times A) \uplus (B \times B)$.

while (3) in the category of finite-dimensional vector spaces becomes

Theorem 2. Let $V, W$ be finite-dimensional vector spaces. Then there exists an injective linear map from $(V \otimes W) \oplus (W \otimes V)$ to $(V \otimes V) \oplus (W \otimes W)$.

In the category of unitary representations of a compact (or finite) group $G$, a little character theory (or complete reducibility together with Schur's lemma) allows one to similarly interpret (3') in this category:

~~.~~Theorem 3. Let $V, W$ be finite-dimensional unitary representations of a compact group $G$. Then there exists an injective $G$-equivariant linear map from $(V \otimes W) \oplus (W \otimes V)$ to $(V \otimes V) \oplus (W \otimes W)$

Theorem 3. Let $V, W$ be finite-dimensional unitary representations of a compact group $G$. Then there exists an injective linear map from $\operatorname{Hom}_G(V,W) \times \operatorname{Hom}_G(W,V)$ to $\operatorname{Hom}_G(V,V) \times \operatorname{Hom}_G(W,W)$.

One can presumably state similar theorems for modules of semisimple algebras over an algebraically closed field, or for various types of vector bundles (or finite covers) over a fixed base space, although I won't attempt to do so here.

Anyway, these results suggest that there may be a way to categorify the inequalities (1), (3), (1'), (3') and/or the identities (2), (2'), for instance by finding a bijective proof (or perhaps an "injective proof") of Theorem 1. However, despite the simplicity of these inequalities and identities this seems to be a surprisingly difficult task. In the ordered case in which one knows that $a=b+k$ or $b=a+k$ for some natural number $k$, one can interpret $(a-b)^2$ as $k^2$, so it is not difficult to categorify Theorem 1 if one possesses an injection from $A$ to $B$ or vice versa, and similarly for Theorem 2 and Theorem 3. From this and the trichotomy of order one can finish off Theorem 1 or Theorem 2, though this is an argument which requires one to make a number of arbitrary choices (as the injections here are not canonical) and so one might consider this to be an incomplete categorification. And in any event, this trick does not seem to recover the full strength of Theorem 3, since there need not be an injective $G$-equivariant map from $V$ to $W$ or vice versa. So I'm wondering if there is another way to categorify these results? For the vector space results (Theorem 2 and Theorem 3) it seems natural to try to use K-theory somehow to achieve this goal (since K-theory already has a formalism for taking formal differences of vector spaces), but I don't know enough K-theory to take this idea further.

[Closely related questions to these were discussed some years ago at the n-category cafe here and here regarding the categorification of the Cauchy-Schwarz inequality, but the results of the discussion were inconclusive, although David Speyer did show that there was an obstruction to categorifying Theorem 3 in the case $G = Z/2Z$ in that the injection could not be natural.]

thatimportant: indefinite products also matter much. So unless you start with something like Hilbert spaces, any good generalizations look unreasonable. – Anton Fetisov May 23 '13 at 18:50